Symplectic representation of higher-order guiding-center theory

Two representations of guiding-center theory are possible depending on whether the guiding-center Poisson bracket (i.e., the symplectic structure) or the Hamiltonian contains higher-order corrections due to the nonuniformity of the magnetic field. By combining the guiding-center parallel hierarchy with the symplectic representation, the guiding-center equations of motion are derived with second-order corrections included in the symplectic structure without the need of carrying out the guiding-center transformation to second order. Guiding-center polarization and magnetization are thus shown to arise naturally from higher-order guiding-center theory within the context of a two-step derivation of nonlinear gyrokinetic theory.\footnote{A.~J.~Brizard and T.~S.~Hahm, Rev.~Mod.~Phys.~{\bf 79}, 421 (2007).}